Robust optimization based EV charging

Alireza Soroudi and Andrew Keane School of Electrical and Computer Engineering University College Dublin, Dublin , Ireland.

Email: Alireza.soroudi@ucd.ie Andrew.keane@ucd.ie

Abstract-With the introduction of new technologies like electric vehicles and smart grids the operation and planning of power systems are subject to major changes. These technologies can bring various ftexibilities to different entities involved in decision making. This paper proposes a robust optimization based method to optimal charging/discharging of electric vehicles con­sidering the electricity price uncertainties. The objective function is defined as the total operating costs of energy procurement in distribution networks which is tried to be minimized while considering the technical constraints of the problem.

I. IN TRODUC TION

The new paradigm of holistic transportation and power system management has attracted a great deal of attention specially with the advent of smart grids. The smart grid is defined as an electric grid in which there exists bidirectional data/command exchange between market players [1]. This is done in order to ensure the reliability and security of energy transition from supply section to end users. The innovative technologies like electric vehicles have the potential to fa­cilitate this task. The key point is that how to capture these potential benefits and promote their role in future. The electric vehicle (EV) can play different roles such as load, generation and energy storage unit [2]. Different energy management and control strategies have been proposed in the literature to capture the potential flexibilities that electric vehicle (EV) can bring to different power system entities as shown in Fig.l.

Fig. 1. The potential flexibilities of electric vehicles

These flexibilities range from end users electricity bill reduction to distribution and transmission impact assessment by operators and utilities. This paper, would provide a model to optimally schedule the EV s in an uncertain electricity market in order to minimize the total energy payments.

978-1-4799-6075-0/14/$31.00 ©2014 IEEE

II. LITERATURE REVIEW

Different approaches have been proposed in the literature to facilitate the flexibility provision by EV to the electric grids. The physical nature of power system requires the instanta­neous balance between the demand and generation. With the increasing share of renewable energy in energy procurement procedure, the importance of flexibility services has become more vital. These services play a key role in maintaining the balance between demand and supply side. The EV s have been identified as an interesting flexibility service provider in different circumstances such as :

• Spinning reserve supply

• Provision of system regulation

• Valley filling and target load following

• Household energy management [3]

• Shaving the peak of the load curve [4]

• Congestion Prevention in the Distribution networks

• Offering secondary reserve by a pool of EV s

• Releasing spinning reserve to electric systems

• Facilitating the load shifting and shaving

• Allowing more flexibility in setting the tradeoff be-tween cost and user comfort

• Load shifting potential of electric vehicles

• Minimizing losses and improve voltage profile [5], [6]

• Possible positive impact on life duration of a trans­former supplying a residential area

III. PROBLEM FORMULATION

The optimal decisions on energy procurement strategy should be taken at each stage to not only reduce the total payments but also not violating the technical constraints. The following optimization problem is solved:

m�nz = L (!;tAt tEWT

F(fl,II)::; 0 G(fl, II) = 0

(1)

(2) (3)

(!;t in (1) is the hourly energy bought from the upstream network. fl and II represent the decision variables and input parameters (price values and technical data), respectively. T is the operating horizon. At is the uncertain electricity price at

time t in day ahead market. F and G represent the inequality and equality constraints of the optimization framework as de­scribed in (6) to (19), respectively. In this paper, a new strategy is proposed that tries to minimize total payments related to the active power procurement. Obviously, the optimal actions 1) directly depend on the input parameters (II) including price values for the day ahead market. The issue is that usually there is limited information about the electricity prices of the next day. The optimization problem can therefore be formulated as follows:

�nz = L CEtAt (4) tE1jJr

Amin < A < Amax t - t - t (5) (6)ta(19)

Af'in, Af'ax are the lower and upper bounds of At, respec­tively. It is assumed that no information is available from day ahead market prices other than these bounds. The power flow equations to be satisfied Vi E \[In, Vt E \[IT, V £ E \[I L are:

Qnet=Vit '" Y;·Vtsin(8·t-8·t-O··) �,t 1" L..J 2J ], 1" ], 2J

jEWn Vmin ::::; Vi,t ::::; Vmax Ii,t = Y£=ij (Vi,t - Vj,t) ::::; Ii

(6)

(7) (8) (9)

(10)

(11) (12)

where prr, Qi'f/ in (7) and (8) are the net injected active and reactive power to bus i, respectively. lij, Oij are the magnitude and angle of the admittance connecting bus i to j, respectively. Vi,t, Vmin, Vmax in (11) are the voltage magnitude, minimax operating limits of each bus, respectively. If in (12) is the current passing through feeder £ and If in (12) is the maximum allowable current in feeder £. pFt, Qft in (7) and (8) are the active and reactive power injected to 'the network by the DG units or grid connection. �i,v is a binary parameter indication if vehicle v is connected to bus i (�i,v = 1) or not (�i,v =

0). \[In, \[IT, \[I L are the set of system nodes, operating hours, feeders, respectively. Ptc�/dch is the charged/discharged power of EVs in (7). '

The operation modeling ofEV is described in (13) to (19).

c ch Pt��h tr SOCt,v = SOCt-l,v + "lvPt,v - TJ� -Pt,v

ini c ch Pt��h tr SOCt v = Ev + TJvPt v - -d- - Pt v , ' �v ' SOC':;"in ::::; SOCt,v ::::; Soc':;"ax Ptt,: = tlDt,vEv pch,vUch < pch < pch,vUch m'l,n t,v - t,v _ max t,v pdch,vUdch < pdch < pdch,vUdch m'l,n t,v - t,v _ max t,v U�� + Ut��h ::::; 1

(13)

(14) (15) (16) (17) (18) (19)

The state of charge in vth EV at time t (Bact,v) depends on the state of charge at time t - 1 (BaCt-l,v) as well as the charging/discharging or traveling state of the EV as modeled in (13) and (14). The relation between the required energy

for traveling of vth EV (Pi:) depends on the traveling distance (f).Dt,v) and also the efficiency of the vehicle (Ev) as described in (16). The state of charge should be kept between operating limits as (15). The charging and discharging rate of each EV are limited by technical characteristics as well as the operating state as enforced by (17) and (18). It is assumed that each EV is either in charging Uf� = 1 / discharging state Udch = 1 or traveling state uch + Udch = 0 as described in t,v t,v t,v (19).

IV. PROPOSED STRATEGY

One of the most important issues of energy management systems is risk management. The term "risk" has various defi­nition. In financial analysis, it may be defined as a loss in profit or excess in a cost function. In technical analysis, it may be defined as a contingency, voltage instability, overloading a line, over voltage or under voltage phenomena. Generally speaking, risk is defined as any undesired event due to the occurrence of unpredicted issue. Different uncertain parameters may lead to risk. These uncertain parameters are usually the output power of renewable energy resources such as wind turbines or photo voltaic cells, demand uncertainty, equipment failures and competitive behavior of electricity market participants [7]. The risk hedging tools are used in order to mitigate the technical or financial risks [8]. Different models and techniques can be found in the literature which propose risk hedging approaches. Some of these models are as follows: stochastic techniques [9], [10], fuzzy methods [11], Information Gap Decision Theory (IGDT) [12] and robust optimization [13], [14]. This paper uses the robust optimization technique to model the uncertainties associated with electricity prices. The robust optimization concept can be used to minimize z in eq. (4) without knowing the exact values of At. Additionally, the optimal decision making should be done in a way that these actions still remain optimal even though the actual values (An of uncertain parameters deviate (to some degree r) from the forecasted values A{. Two cases may happen: first, the actual price Af is more than the forecasted price A{. The constraint for uncertainty modelling of the price can be expressed as:

A� = At + tli';t tli = Af'ax -At o ::::; ';t ::::; 1

where, �t is the prediction error.

(20a) (20b) (20c)

Actually the main concern of the decision maker is on (20a) , (20b) where the actual prices may be more than the forecasted values. Thus, the formulation expressed in (4) , (5) can be replaced by the following one:

m�nz = L (!;tA{ + (!;ttli';t tET

o ::::; ';t ::::; 1 L';t::::; r tET

(6)to(19)

(21a)

(2Ib) (2Ic)

r in (21c) is the maximum total deviation that can be tolerated. It varies from 0 (no prediction error) to 24 (100% prediction error) (increases with the conservativeness of the decision maker). For example, if r = 4 this means that the algorithm will remain robust even though the maximum total prediction

error is 100% in 4 hours or 50% in 8 hours of the day ahead market. The robust counterpart of (21) is represented as [15]:

. '" '" At {max�t Q;tl:li t;t } �nz = L....- "'t t + (21b), (21c) tET ( 6)to(19)

(22)

The formulation described in (22), requires to solve a bi-level optimization since the inner maximization tries to simulate the worst case realization of uncertain price (by changing �t) while the outer minimization attempts to decrease the undesired impacts of uncertain prices by controlling �. According to the duality theory [14], (22 ) is equivalent to (23) as follows:

m�nz = A{Q;t + L(t + IT tET

Y + (t � (A;"ax - A{)Q;t Y, (t � 0 (6)to(19)

where (i, Yare dual variables.

(23a)

(23b) (23c)

It is obvious that the resulted single level optimization is easier to solve than the original bi-level optimization structure. The decision variables (�), parameters (II) and the sets are as follows:

(24)

(25)

(26)

V. SIMULATION RESULTS

The proposed robust optimization based model is examined on a 33-bus distribution system. The data of this system including the data of loads, and feeders are given in [16]. In order to better explore the benefits of electric vehicles on distribution networks, the load level is decreased by 50%. The proposed algorithm is implemented in GAMS environment solved by SNOPT solver running on an Intel®Xeon ™CPU E5-l620 3.6 GHz PC with 8 GB RAM. It is assumed that there are only 10 EV s in the system under study. The technical data regarding the EV s is given in Table I.

TABLE I. ELECTRIC VEHICLE CHARACTERISTICS DATA Parameter Value Unit

ry� 93 % ry� 90 % E�m 3 kWh Soc�ax 40 kWh Soc�in 1 kWh pc�: 0 kW p::::,� 20 kW pd".�,v 0 kW p���v 20 kW Ev � kW/km

The traveling patterns of EVs are given in Table II. It is assumed that the required energy for demand supply is the pool market. The purchased power from pool market is injected to the network through slack bus which is bus i = 1. The electric demand and electricity price patterns are shown in Fig. 2.

TABLE II. ELECTRIC VEHICLE TRAVELING PATTERN (KM) AND CONNECTED NODES

Time v, v, v, v, v, v, v, v, v, Vw " 0 0 0 4.6 0 0 0 4 0 0 t2 0 0 0 1.8 0 2 0 0 0 0 t, 0 5 0 0 0 2 0 0 0 2 t4 0 0 0 0 0 0 0 1 0 0 t, 2.4 0 0 0 0 4 2.8 3 0 4 t, 0 4.8 0 0 0 1 0 0 0 0 t, 0 0 0 0 0 0 0 0 0 0 t, 4.8 0 1 2 0 0 0 0 0 2 t, 0 0 0 0 1 0 0 0 0 1 tw 0 2.4 0 0 0 0 0 0 0 1 tn 0 0 0 4.6 2 0 0 4.4 0 0 t" 4 0 0 0 4 3 0 0 0 0 tn 0 0 0 0 0 0 0 0 4.2 0 t" 0 0 0 0 0 0 0 4 0 0 t" 0 0 0 0 0 0 0 0 3.8 0 t" 3.6 0 0 4.6 0 0 0 0 0 4 t" 0 0 3.6 0 0 0 0 3 0 4 t" 0 0 0 4 0 0 0 1.8 0 0 t" 0 0 4 0 0 2.6 0 0 2 4 t" 0 0 0 0 0 0 4.2 3.2 2.2 0 t" 0 0 4.8 3.8 0 0 0 2.6 1 1 t" 0 0 0 0 0 0 0 0 0 0 t23 0 4.8 0 0 0 0 0 0 0 2.2 t,. 0 0 0 0 0 0 3.2 0 0 0

i: "'i,v 1 12 20 23 21 32 15 28 17 22 33

f'OO�1 m . . . . . . . . . . . . . . . � 90 - : - : --: - :- -:- -: - ' :- - :- -: - :-- � -, " - -.- -: - .-_ .

I 00 � " •• " •• '.': •. ' ••...••.

...• • ' .. : •....••...••

....••.. : •

....•..••

...•• ': •

. ' •• ...••

.••••••.•..•• ••••.•.. • ••• •.•.. •• •• • •.. •. ••• •

.• .• i. 70 -: - : '_ . - ,. , ,: , . , :, ,., ,: _ ., - : -: -: - ,. - -" , . , , : -

• - . - - . _ - .-� 60 - -: -; - - .: -. : . - .:. -.:. -.:. -.:. -.:. -.: - -:. - - � -- : -- : --.: --.: .. : . . : E . . . . . . . . . . . . . . . . . . .!!: 501 2 3 4 5 6 7 8 9101112131 415161718192021 22 2324

Hour(!)

Fig. 2. The electric demand and electricity price patterns

Two case studies have been performed to demonstrate the effectiveness of the proposed methodology namely, random case and centrally controlled charging.

A. Random charging case In this case, it is assumed that the EV owners perform

the charging and do not inject any energy to the grid (e.g. pl�h = 0). The charging pattern is completely random and do not consider the price (as well as its uncertainties). The total energy charged (Ptch) in random case is depicted in Fig. 3.

1 2 3 4 5 6 7 8 9 10 11 12 13141516171819202122 2324 Time

Fig. 3. The total energy charged (Ptch) in random case

The total Eoct in random charge case is shown in Fig. 4.

1 2 3 4 5 6 7 8 9 1 011 1 2 13141516171 81 9202122 2324 Time

Fig. 4. The total Eoct in random charge case

The hourly energy losses vs f in random case are shown in Fig. 5. The total energy losses are 0.51625 MWh.

1 0.02 5

I >. 0.02

i � s: 0.015 :c

1 2 3 4 5 6 7 8 9 10 1112 1314151617181920 2122 2324 Time

Fig. 5. The hourly energy losses vs r in random case

The total payments depend on price uncertainty. As the price uncertainty increases, the total payments would increase. The total energy payments compared are shown in Fig. 11.

B. Centrally controlled charging/discharging case In this case, it is assumed that the charging/discharging of

EVs is performed centrally. The charging/discharging pattern is optimally determined with considering the electricity price uncertainties.

The total energy charged (p{h) in centrally controlled case is depicted in Fig. 6. The total energy discharged (ptdch) in

Time

Fig. 6. The total energy charged (p{h) in controlled case

controlled case is shown in Fig. 7. The total state of charge vs

Fig. 7. The total energy discharged (Ptdch) in controlled case

f in centrally controlled case is shown in Fig. 8.

Fig. 8. The state of charge vs r in centrally controlled case

The charging and discharging pattern of EV s directly affects the active losses in distribution network. This impact is investigated in this case. The hourly energy losses vs f in controlled case are given in Fig. 9. The total energy losses

� !. 0.03 Ii .2 0.02 � 51 0.01 � � 0

24

Fig. 9. The hourly energy losses vs r in controlled case

vs f in controlled case is shown in Fig. 10. The total energy payments in both cases are compared in Fig. 11.

Comparing the payments in both cases we observe that, the centrally controlled case results in lower total operating costs in every budget of uncertainty (f).

If the percentage of EV loads of the the total load in distribution network is increased, then the energy payments

0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324 r

Fig. 10. The total energy losses vs r in controlled case

10 0!--'-1 �2 -':3--'

4:-5�0 �7 -':O--'

9:-:1"=-0 1:':-1-:':12-='13:-'14'"'1"=-51:':-0-:':17-='10:-'19=-=2"=-0 2:':-1-:':22-='23:-'2 4'-­r

Fig. II. The total energy payments compared in both cases

reduction would be more significant. Uncertainty modeling is important because it would enable the DSO to avoid the risk of high values of electricity prices. The level of conservatism can be adjusted by choosing the appropriate (r) level. The proposed formulation is computationally attractive. The un­certainty modeling of electricity prices has a linear structure and does not add more complexity to the existing model.

VI. DISCUSSIONS

Considering the operating strategy carried out, the general conclusions below are in order:

1) The proposed framework can be extended to consider the impacts of any renewable distributed generation (DG) units existing in the network.

2) The existing model can be enhanced if it is integrated with (Demand Response) DR programs.

3) Other uncertainties can be considered in the model to get closer to the reality. These uncertainties may be related to the demand, generation of renewable power generations or conventional technologies.

4) Other uncertainty modeling techniques can be used to deal with uncertainties such as Information Gap Decision Theory (IGDT). The IGDT technique can be used to model opportunistic behavior of decision maker (DSO).

5) Using the information provided by smart grid facilities, the price intervals can be updated. This means it is not necessary to run the model on a day ahead basis. The rolling window can move from tl and updates the available information about the uncertain parameters.

6) The distribution network reconfiguration can also be regarded as an option for reducing the operating costs.

7) The capacitor switching can also be used to reduce the operating costs.

VII. CONCLUSION

An optimization based technique was proposed to obtain the optimal scheduling of the EV s. The robust optimization approach is used to deal with the uncertainty of electricity prices. The electric vehicles have been used as the energy storage device in order to maximize the flexibility of distribu­tion network operator. The obtained results from the proposed risk-averse strategy ensure the decision maker that although the predicted values of the uncertain input parameters are not exact but the outcome of the proposed model (payments) would be immune against the prediction error to some controlled extent (r).

ACKNOWLEDGMENT

The work of A. Soroudi was conducted in the Electricity Research Centre, University College Dublin, Ireland, which is supported by the Commission for Energy Regulation, Bord Gais Energy, Bord na Mona Energy, Cylon Controls, Eir­Grid, Electric Ireland, EPRI, ESB International, ESB Net­works, Gaelectric, Intel, SSE Renewables, UTRC and Viridian Power & Energy. A. Soroudi is funded through Science Foundation Ireland (SFI) SEES Cluster under grant number SFI/09/SRCIE1780. A. Keane is supported by the SFI Charles Parsons Energy Research Awards SFI/06/CPIE005.

REFERENCES

[I] P. Siano, "Demand response and smart gridsa survey;' Renewable and Sustainable Energy Reviews, vol. 30, pp. 461-478, 2014.

[2] M. Falvo, G. Graditi, and P. Siano, "Electric vehicles integration in demand response programs," in Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), 2014 International Symposium on, June 2014, pp. 548-553.

[3] A. Dargahi, S. Ploix, A. Soroudi, and F. Wurtz, "Optimal household en­ergy management using v2h flexibilities," COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 33, no. 3, pp. 777-792, 2014.

[4] V. Calderaro, V. Galdi, G. Graber, G. Graditi, and F. Lamberti, "Impact assessment of energy storage and electric vehicles on smart grids," in Electric Power Quality and Supply Reliability Conference (PQ), 2014, June 2014, pp. 15-18.

[5] A. O'Connell, D. Flynn, and A. Keane, "Rolling multi-period optimiza­tion to control electric vehicle charging in distribution networks," IEEE Trans. Power Syst., vol. 29, no. 1, pp. 340-348,2014.

[6] P. Richardson, D. Flynn, and A. Keane, "Optimal charging of electric vehicles in low-voltage distribution systems," Power Systems, IEEE Transactions on, vol. 27, no. 1, pp. 268-279, Feb 2012.

[7] P. Lamaina, D. Sarno, P. Siano, A. Zakariazadeh, and R. Romano, "A model for wind turbines placement within a distribution network acquisition market," Industrial Informatics, IEEE Transactions on, vol. PP, no. 99, pp. 1-1,2014.

[8] M. A. Tajeddini, A. Rahimi-Kian, and A. Soroudi, "Risk averse optimal operation of a virtual power plant using two stage stochastic program­ming;' Energy, vol. 73, no. 14, pp. 958-967, 2014.

[9] A. Soroudi, "Possibilistic-scenario model for dg impact assessment on distribution networks in an uncertain environment;' IEEE Transactions on Power Systems" vol. 27, no. 3, pp. 1283-1293,2012.

[10] T. Amraee, A. Soroudi, and A. Ranjbar, "Probabilistic determination of pilot points for zonal voltage control," Generation, Transmission Distribution, lET, vol. 6, no. 1, pp. 1-10, January 2012.

[11] A. Soroudi, M. Ehsan, R. Caire, and N. Hadjsaid, "Possibilistic evalua­tion of distributed generations impacts on distribution networks;' IEEE Transactions on Power Systems" vol. 26, no. 4, pp. 2293-2301, 2011.

[12] A. Soroudi and M. Ehsan, "Igdt based robust decision making tool for dnos in load procurement under severe uncertainty," Smart Grid, IEEE Transactions on, vol. 4, no. 2, pp. 886-895, 2013.

[13] A. Soroudi, "Smart self-scheduling of gencos with thermal and energy storage units under price uncertainty," International Transactions on Electrical Energy Systems, vol. 24, no. 10, pp. 1401-1418,2014.

[14] A. Soroudi and T. Amraee, "Decision making under uncertainty in energy systems: State of the art," Renewable and Sustainable Energy Reviews, vol. 28, pp. 376-384, 2013.

[15] A. Soroudi, "Robust optimization based self scheduling of hydro­thermal genco in smart grids," Energy, vol. 61, pp. 262-271, 2013.

[16] M. Baran and F. Wu, "Network reconfiguration in distribution systems for loss reduction and load balancing," Power Delivery, IEEE Transac­tions on, vol. 4, no. 2, pp. 1401-1407, Apr 1989.